How to remove part of a model and create a handle?
Not a direct answer, but I propose this approach :




Upper left, the base shape (only extruded cube)
Lower left, the same with a subsurf modifier
Upper right, vertices adjustment in order to fit the shape
Lower right, the same with subsurf




Edit for some explanations as Thefundude said to be new to Blender :

The main idea is to start from a basic shape which as the topology of the bottle. So mainly here, has a hole below the handle and is opened at the top.

The very basic first shape is here for that (defining the topology), but also, as it is very basic (few vertices), its vertices can be used as handles to reshape this starting mesh to the wanted shape.

The subsurface modifier is used to round the shape.

Here is how to set it up :


Select your starting mesh
Go to the modifier panel and select the subsurface modifier
Give it one or two subdivisions




Now the first part to enhance will probably be the handle. We want it curve. We will use the spin tool for that.


Remove the non curve part
Place your cursor at a pivot point (the spin uses the cursor as pivot). Select the location and Shift+S and 'cursor to selected'
Select the vertices to spin
Place the view in the spin axis. Here front view so NumPad1
Then Alt+R to invoke the spin tool
Change the amount of steps to an amount which more or less preserves the proportionalities of the whole model (the same precision everywhere, for now)
Finally, as the spin has created vertices which overlaps already created vertices (these on the bottom right here), select them (or all) and remove doubles.


 

After that, this is mainly tuning and adjusting matter. 

For instance you can scale the handle :



Or add some more geometry to refine the shape :



Or want to add more edge loops, in particular to flatten the base (I used here the 'inset tool' which is invoked by I (then adjust moving the mouse) :



Step by step (but starting from an appropriated topology, adding few vertices as you want to master the shape easily) you can refine your shape without arriving to the complex situation of your question.

Cheers !